Finite Group Databasebeta

← C22×C4×Q8⋊C9 #11919677

Subgroups — C22×C4×Q8⋊C9 #11919677

Order 1152.

893 conjugacy classes of subgroups; only the 317 normal subgroups are shown.

normal subgroups only

This group has 893 conjugacy classes of subgroups — too many to draw. Only its 317 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×Q8⋊C9C211yesmaximal, characteristic
2C2×C4×Q8⋊C9C216yesmaximal, direct product factor
3C22×C3×C4×Q8C311yesmaximal, characteristic
4C23×C4×C9—41nomaximal
5C22×Q8⋊C9C414yesdirect product factor
6C22×Q8⋊C9C2213yes—
7C4×Q8⋊C9C2214yesdirect product factor
8C23×C3×Q8C2×C311yescharacteristic
9C2×C3×C4×Q8—36no—
10C2×C3×C4×Q8C2×C316yes—
11C22×C3×C4⋊C4—31no—
12C22×C3×C42—31no—
13C24×C9—41no—
14C22×C4×C9—48no—
15C22×C4×C9—46no—
16C2×Q8⋊C9C2×C416yesdirect product factor
17C2×Q8⋊C9C2311yescharacteristic
18C22×C4×Q8C911yescharacteristic, complemented, Sylow
19C22×C3×Q8—34no—
20C22×C3×Q8—33no—
21C22×C3×Q8C3×C414yes—
22C22×C3×Q8C22×C313yes—
23C23×C3×C4—31no—
24C23×C3×C4—31no—
25C23×C3×C4A411yescharacteristic
26C3×C4×Q8—38no—
27C3×C4×Q8—312no—
28C3×C4×Q8C22×C314yes—
29C2×C3×C4⋊C4—36no—
30C2×C3×C4⋊C4—36no—
31C2×C3×C42—36no—
32C2×C3×C42—36no—
33C23×C9—44no—
34C23×C9—44no—
35C23×C9—44no—
36C23×C9—43no—
37C2×C4×C9—424no—
38C2×C4×C9—44no—
39Q8⋊C9C22×C411yescharacteristic, direct product factor
40C23×Q8C2×C911yescharacteristic
41C2×C4×Q8—36no—
42C2×C4×Q8C2×C916yescomplemented
43C22×C4⋊C4—31no—
44C22×C42—31no—
45C24×C3C2×A411yescharacteristic
46C2×C3×Q8—32no—
47C2×C3×Q8—312no—
48C2×C3×Q8—33no—
49C2×C3×Q8—312no—
50C2×C3×Q8—36no—
51C2×C3×Q8C2×C3×C416yes—
52C2×C3×Q8C23×C311yescharacteristic
53C22×C3×C4—34no—
54C22×C3×C4—34no—
55C22×C3×C4—33no—
56C22×C3×C4—33no—
57C22×C3×C4—36no—
58C22×C3×C4—64no—
59C22×C3×C4SL(2,𝔽3)18yes—
60C22×C3×C4C2×A416yes—
61C3×C4⋊C4—34no—
62C3×C4⋊C4—312no—
63C3×C42—312no—
64C3×C42—34no—
65C22×C9—416no—
66C22×C9—46no—
67C22×C9—46no—
68C22×C9—46no—
69C22×C9—41no—
70C4×C9—48no—
71C22×Q8—33no—
72C22×Q8—34no—
73C22×Q8C4×C914yescomplemented
74C22×Q8C22×C913yes—
75C23×C4—31no—
76C23×C4—31no—
77C23×C4C22⋊C911yescharacteristic
78C4×Q8—38no—
79C4×Q8—312no—
80C4×Q8C22×C914yescomplemented
81C2×C4⋊C4—36no—
82C2×C4⋊C4—36no—
83C2×C42—36no—
84C2×C42—36no—
85C23×C3C4×A414yes—
86C23×C3C2×SL(2,𝔽3)14yes—
87C23×C3#41⋊C314yes—
88C23×C3C22×A413yes—
89C3×Q8—36no—
90C3×Q8—31no—
91C3×Q8—38no—
92C3×Q8—36no—
93C3×Q8C22×C3×C411yescharacteristic
94C2×C3×C4—36no—
95C2×C3×C4—312no—
96C2×C3×C4—36no—
97C2×C3×C4—33no—
98C2×C3×C4—31no—
99C2×C3×C4—34no—
100C2×C3×C4—612no—
101C2×C3×C4C2×SL(2,𝔽3)124yes—
102C2×C3×C4C22×A414yes—
103C2×C9—41no—
104C2×C9—41no—
105C2×C9—41no—
106C2×C9—412no—
107C24C2×C22⋊C911yescharacteristic
108C2×Q8—32no—
109C2×Q8—312no—
110C2×Q8—33no—
111C2×Q8—312no—
112C2×Q8—36no—
113C2×Q8C2×C4×C916yescomplemented
114C2×Q8C23×C911yescharacteristic
115C22×C4—34no—
116C22×C4—34no—
117C22×C4—33no—
118C22×C4—33no—
119C22×C4—36no—
120C22×C4—64no—
121C22×C4Q8⋊C918yesdirect product factor
122C22×C4C2×C22⋊C916yes—
123C4⋊C4—34no—
124C4⋊C4—312no—
125C42—34no—
126C42—312no—
127C22×C3C4×SL(2,𝔽3)116yes—
128C22×C3C2×C4×A416yes—
129C22×C3C22×SL(2,𝔽3)16yes—
130C22×C3C2×#41⋊C316yes—
131C22×C3C23×A411yescharacteristic
132C3×C4—31no—
133C3×C4—36no—
134C3×C4—31no—
135C3×C4—64no—
136C3×C4C22×SL(2,𝔽3)18yes—
137C9—41noSylow
138C23C4×C22⋊C914yes—
139C23C2×Q8⋊C914yes—
140C23#41⋊C914yes—
141C23C22×C22⋊C913yes—
142Q8—36no—
143Q8—31no—
144Q8—38no—
145Q8—36no—
146Q8C22×C4×C911yescharacteristic, complemented
147C2×C4—31no—
148C2×C4—33no—
149C2×C4—36no—
150C2×C4—312no—
151C2×C4—36no—
152C2×C4—34no—
153C2×C4—612no—
154C2×C4C2×Q8⋊C9124yesdirect product factor
155C2×C4C22×C22⋊C914yes—
156C2×C3C2×C4×SL(2,𝔽3)112yes—
157C2×C3C22×C4×A411yescharacteristic
158C2×C3C23×SL(2,𝔽3)11yescharacteristic
159C2×C3C22×#41⋊C311yescharacteristic
160C22C4×Q8⋊C9116yesdirect product factor
161C22C2×C4×C22⋊C916yes—
162C22C22×Q8⋊C916yes—
163C22C2×#41⋊C916yes—
164C22C23×C22⋊C911yescharacteristic
165C4—31no—
166C4—31no—
167C4—36no—
168C4—64no—
169C4C22×Q8⋊C918yesdirect product factor
170C3C22×C4×SL(2,𝔽3)11yescharacteristic
171C2C2×C4×Q8⋊C9112yesdirect product factor
172C2C22×C4×C22⋊C911yescharacteristic
173C2C23×Q8⋊C911yescharacteristic
174C2C22×#41⋊C911yescharacteristic
subgroupG/H
center, hypercenterC23×C3×C4A4
commutator, hypocenterQ8C22×C4×C9
FrattiniC22×C3C23×A4
FittingC22×C3×C4×Q8C3
socle, prime radicalC24×C3C2×A4
cosocleC2×C3×Q8C23×C3
LCSG ▷ Q8
UCSC1 ◁ C23×C3×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C4×Q8⋊C9C2direct (N × H)
5. C22×Q8⋊C9C4direct (N × H)
7. C4×Q8⋊C9C22direct (N × H)
16. C2×Q8⋊C9C2×C4direct (N × H)
18. C22×C4×Q8C9semidirect
39. Q8⋊C9C22×C4direct (N × H)
42. C2×C4×Q8C2×C9semidirect
73. C22×Q8C4×C9semidirect
80. C4×Q8C22×C9semidirect
113. C2×Q8C2×C4×C9semidirect
121. C22×C4Q8⋊C9direct (N × H)
146. Q8C22×C4×C9semidirect
154. C2×C4C2×Q8⋊C9direct (N × H)
160. C22C4×Q8⋊C9direct (N × H)
169. C4C22×Q8⋊C9direct (N × H)
171. C2C2×C4×Q8⋊C9direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×Q8⋊C92
C2×C4×Q8⋊C92
C22×C3×C4×Q83
C23×C4×C94